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Phenomenological Equations of State for the Quark-Gluon Plasma

Peter N. Meisinger, Travis R. Miller, Michael C. Ogilvie

TL;DR

This paper tackles the quark-gluon plasma equation of state for SU(N) gauge theories by proposing two simple, one-parameter phenomenological models (Model A with a massive gluon and Model B with a color-neutrality scale) that incorporate center symmetry and the full Polyakov loop eigenvalue spectrum. The central idea is that thermodynamics are governed by the distribution of Polyakov loop eigenvalues, with confinement arising from eigenvalue repulsion and deconfinement driven by eigenvalue rotation toward the center at high temperature; both models reproduce lattice-like behavior in the range $T_d$ to about $5T_d$ and predict the correct order of the deconfinement transition (second order for $N=2$, first order for $N=3,4,5$) and a smooth large-$N$ limit. Quarks are included perturbatively, showing heavy-quark effects shift $T_d$ slightly and light quarks can turn the transition into a crossover, depending on $N_f$, while the framework remains applicable across the relevant temperature range. The work emphasizes that accurate modeling of the EoS in the QGP requires eigenvalue-based descriptions beyond the trace of the Polyakov loop and lays groundwork for integrating with other approaches (e.g., HTL) and with chiral dynamics in future studies.

Abstract

Two phenomenological models describing an SU(N) quark-gluon plasma are presented. The first is obtained from high temperature expansions of the free energy of a massive gluon, while the second is derived by demanding color neutrality over a certain length scale. Each model has a single free parameter, exhibits behavior similar to lattice simulations over the range T_d - 5T_d, and has the correct blackbody behavior for large temperatures. The N = 2 deconfinement transition is second order in both models, while N = 3,4, and 5 are first order. Both models appear to have a smooth large-N limit. For N >= 4, it is shown that the trace of the Polyakov loop is insufficient to characterize the phase structure; the free energy is best described using the eigenvalues of the Polyakov loop. In both models, the confined phase is characterized by a mutual repulsion of Polyakov loop eigenvalues that makes the Polyakov loop expectation value zero. In the deconfined phase, the rotation of the eigenvalues in the complex plane towards 1 is responsible for the approach to the blackbody limit over the range T_d - 5T_d. The addition of massless quarks in SU(3) breaks Z(3) symmetry weakly and eliminates the deconfining phase transition. In contrast, a first-order phase transition persists with sufficiently heavy quarks.

Phenomenological Equations of State for the Quark-Gluon Plasma

TL;DR

This paper tackles the quark-gluon plasma equation of state for SU(N) gauge theories by proposing two simple, one-parameter phenomenological models (Model A with a massive gluon and Model B with a color-neutrality scale) that incorporate center symmetry and the full Polyakov loop eigenvalue spectrum. The central idea is that thermodynamics are governed by the distribution of Polyakov loop eigenvalues, with confinement arising from eigenvalue repulsion and deconfinement driven by eigenvalue rotation toward the center at high temperature; both models reproduce lattice-like behavior in the range to about and predict the correct order of the deconfinement transition (second order for , first order for ) and a smooth large- limit. Quarks are included perturbatively, showing heavy-quark effects shift slightly and light quarks can turn the transition into a crossover, depending on , while the framework remains applicable across the relevant temperature range. The work emphasizes that accurate modeling of the EoS in the QGP requires eigenvalue-based descriptions beyond the trace of the Polyakov loop and lays groundwork for integrating with other approaches (e.g., HTL) and with chiral dynamics in future studies.

Abstract

Two phenomenological models describing an SU(N) quark-gluon plasma are presented. The first is obtained from high temperature expansions of the free energy of a massive gluon, while the second is derived by demanding color neutrality over a certain length scale. Each model has a single free parameter, exhibits behavior similar to lattice simulations over the range T_d - 5T_d, and has the correct blackbody behavior for large temperatures. The N = 2 deconfinement transition is second order in both models, while N = 3,4, and 5 are first order. Both models appear to have a smooth large-N limit. For N >= 4, it is shown that the trace of the Polyakov loop is insufficient to characterize the phase structure; the free energy is best described using the eigenvalues of the Polyakov loop. In both models, the confined phase is characterized by a mutual repulsion of Polyakov loop eigenvalues that makes the Polyakov loop expectation value zero. In the deconfined phase, the rotation of the eigenvalues in the complex plane towards 1 is responsible for the approach to the blackbody limit over the range T_d - 5T_d. The addition of massless quarks in SU(3) breaks Z(3) symmetry weakly and eliminates the deconfining phase transition. In contrast, a first-order phase transition persists with sufficiently heavy quarks.

Paper Structure

This paper contains 10 sections, 48 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Perturbative EoS at high T
  3. Failings of the Landau Approach
  4. Model A
  5. Model A for $SU(2)$
  6. Model A for $SU(3)$
  7. Model B
  8. $SU(N)$ Thermodynamics for $N=2,3,4$, and $5$
  9. Quarks
  10. Conclusions

Figures (9)

  • Figure 1: $p/T^4$ versus $T/T_c$ for $SU(3)$
  • Figure 2: $e/T^4$ versus $T/T_c$ for $SU(3)$
  • Figure 3: $\Delta/T^4$ versus $T/T_c$ for $SU(3)$
  • Figure 4: $p/T^4$ versus $T/T_c$ using model A for $N=2,3,4,5$
  • Figure 5: $e/T^4$ versus $T/T_c$ using model A for $N=2,3,4,5$
  • ...and 4 more figures